Lindenbaum-Tarski algebra for the class of Boolean algebras with one distinguished ideal

It is proved that for every n ∈ ω, the quotient with respect to the nth Frechet ideal of the Lindenbaum-Tarski algebra for the class of Boolean algebras with one distinguished ideal is an infinite atomless Boolean algebra. All elements of this algebra having a finite Frechet rank are detailed. A complete description is given for I-algebras whose elementary theory is axiomatizable by a single atorn in some finite quotient with respect to the Frechet ideal of the Lindenbaum-Tarski algebra for the class of Boolean algebras with one distinguished ideal.